4/02/2002 @ 8:00AM

Mandelbrot: A Math Maverick Takes Stock

Benoit
Mandelbrot
Benoit Mandelbrot
, one of the world’s most celebrated mathematicians, believes that our understanding of the stock market is as flawed as medieval astronomy. But the 77-year-old mathematician thinks that he now has the tools to fundamentally revise the way economists envision price variation. The change could be as radical as moving the Earth from the center of the solar system to an orbit round the sun.

More than 40 years ago, Mandelbrot discovered a new kind of math: Instead of accounting for neat and tidy geometry, his new equations described shapes that were rough and jagged, with general patterns repeating in new and unexpected ways. Eventually, he named these new mathematical equations fractals, and in the intervening decades they have been used to describe coastlines, clouds and air turbulence, to create computer graphics and to model the way rocks and sheets of metal wear, fragment and crack.

Now Mandelbrot has brought the fractal toolbox back to the field of price variation, where he made vital contributions decades ago. In the 1960s, while working at
IBM
, Mandelbrot made a stab at using fractals to describe the market; his ideas generated excitement, but not much else. The mathematician, however, still doesn’t like the way many of his colleagues are trying to predict the markets’ ebb and flow–in fact, he thinks they’ve got the wrong idea entirely.

“One would like to predict prices, to predict economic development,” allows Mandelbrot, an imposing man with the attitude and aspect of a polar bear, who splits his time between research labs at IBM and Yale University. “But a preliminary step is just to describe them. It sounds down-to-earth and disappointing, and perhaps too modest, but it is absolutely indispensable.” In Mandelbrot’s opinion, economists have not described the stock market well–so how can they predict it?

The view Mandelbrot is arguing against goes like this: Price is determined by binary decisions, like coin tossing. Heads, the price goes up; tails, it goes down. Investors putting in buy and sell calls should then act a bit like particles in a glass of milk, which remain suspended because they are constantly bustling into one another in a process known as Brownian motion. In theory, lots of tiny, bustling investors should have a similar effect–keeping stocks pretty much suspended, with only slow changes.

Yet the stock market is subject to violent swings. “I have seen cases where something went down from $90 to 90 cents,” Mandelbrot says. Where the model sees a gentle breeze, there is instead a mighty storm. Price drops or runups that the coin-tossing model sees as impossible–changes with a probability of a millionth of a millionth of a millionth of a millionth–occur frequently. “It should never happen,” says Mandelbrot. “Never, never, never. But it happens all the time.”

One solution is to modify the coin-tossing model–to add bells and whistles until the equations match the observations. Mandelbrot thinks this isn’t playing by the rules. “It’s reminiscent of the model they had of the Earth going around the sun in a circle,” he says. “It went in a little circle and then a big circle.” Such endlessly looping orbits were fixed when scientists abandoned the perfect circle for the flattened ellipse, making it possible to track comets and other objects with very elliptical orbits.

The lesson, to Mandelbrot, is obvious: Go with the math that matches the problem, even if it seems strange. In order to model the stock market, he insists, one must use fractals, because they are the only way science has of exactly imagining the rough and uneven nature of a market. “Roughness proved very difficult to handle,” Mandelbrot says, “which is why it was not handled before fractal geometry.” (See “Fractal What? Mandelbrot Explains“)

Mandelbrot’s first foray into fractal economics, when he was discovering ways to use computers to predict fractal systems at IBM in the ’60s, had major impact on the field. But then he hit a wall: Either he could write equations that looked like the stock market but didn’t allow him to predict stock price changes in any meaningful way, or he had a prediction system that did not account for wild price swings.

“I think everyone accepts that his basic point is true,” says
Eugene
Fama
Eugene Fama
, professor of finance at the University of Chicago’s Graduate School of Business. Traditional statistics don’t predict the wild variability of the market very well. But by some measures, Fama says, the market starts too look smooth at larger timescales. “You’re left with a situation where over short intervals his model looks good, but over long intervals it doesn’t seem to work,” Fama says.

Mandelbrot moved on to greener pastures, applying his new mathematical tools to a wide variety of fields. The Fractal Geometry of Nature, a 1982 book on the continual reappearance of fractals in nature, made him famous. Trees, clouds, numerous patterns that we see in daily life–all turned out to be fractal in nature, described by a branch of mathematics that Mandelbrot had started. He became a favorite figure of chaos theoreticians and computer graphics artists, both of whom drew from his work.

Around ten years ago, Mandelbrot says, he began thinking very seriously about how to fix the thorny problems of economics he had attacked so many years before. New techniques developed more than a decade ago allowed him to begin finding ways of measuring the market, of being able to model its angry mood swings with some level of precision. His formula would become a way of taking the market’s temperature, of providing a measurement. The technique is not ready to be used for stock-picking algorithms yet, he says, but he insists it is the proper approach.

“The test of a good theory is that the consequences are numerous and accurate,” Mandelbrot says. “In the case of coin tossing, that is not the case. It is very simple, but it predicts that any day is like any other. My theory says no: If you have a large number of price changes, a small number of days are important.”

“The other days,” he says, “you might as well go to the beach and do nothing.”